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Mención Internacional en el título de doctor SUMMARY, REMARKS AND FUTURE WORK. Culture is one of the classical examples that is normally used to define a complex system. The inherent complexity of humans interacting directly provides enough diversity of phenomena, but if we also consider sociological problems that live in wider timescales, then it becomes boundless. The subject of the interaction is no more the person alone, but the different social constructs that coexist within a society, as, for example, different kind of social norms and expressions. The growing complexity of social systems is also confusing because it has not been traditionally understood in this way, which is logical because we have studied society from the different perspectives that have appeared historically. Humanities have studied the different holistic products that appear from complexity, like literature, philosophy or the different types of artistic studies. As we stated in the introduction, a different perspective appeared when the scientific method started comparative studies of society, bringing new tools, mainly modelling of general effects and statistics. Herbert Spencer popularized the analogy of considering the society as a human being, this is, as a cooperative organization of the different organs and tissues in order to provide a collective structure. And this is the main spirit of complexity sciences, which started to have more importance in the end of the XX century, giving rise to a new wave on how to understand quantitatively social systems. At this moment, we recall the title of this thesis: Behavior and context. The problem in the description of the context is a methodological one. Context is the conjunction of relevant circumstances correlated with the social description of an individual. Traditionally, the social description of a variable has been related with socioeconomic indicators, where the presence of the social structure is only understood with scalar variables. In this work, we show the importance of the network structure in the resemblance of this social profile, and how geometry may act as a proxy of this social variables containing, potentially, more information than just scalar variables. The behavior is correlated with the geometry of this social profile. The importance of the network version of the social profile can be checked by its correlation with the cultural variables of the ego. This is important because it reveals an idiosyncratic manner of making social relationships, and, therefore, the role of social institutions. But the network version is not only a proxy for cultural variables, it is a true identifier of people. A proof of this is the connection between personal profiles and probability of connection, showing that the similarity in the close structural network may be a proxy for social homophily, connected, therefore, to personal and cultural values. Therefore, we have identified a unit component (in terms of measure) of the individual, social and cultural influence on individuals, providing a personal signature around which social homophily appears, even in a complex context where culture and social values play a key role. At the end of the day, this thesis could be summarized by saying that the geometric structure of the close social environment is a signature of the individual and of its social and cultural values. Similarity in this signature is related to social homophily. The other main message that can be extracted from this thesis is related to the importance of ethnic groups in social modelling. Social modelling can not be replicated easily with a markovian model with binary interactions. This is due to the inherent complexity that was pointed out at the beginning of the chapter. Social phenomena exist in several timescales, and because of this they have a lot of implicit information. An ethnic group may exist because of this implicit information, due to, for example, a pre-existing social symbolism; but, in order to arise spontaneously from the social outcome of a binary process (in the way game theory works), some optimization is needed. This is, the existence of a marker does not provide, necessarily, a social separation depending on that variable, if this variable does not provide some kind of improvement in the fitness. This advantage may exist due to an asymmetrical social configuration, for example, in the case of a hierarchy. Previous works have shown the spontaneous appearance of this kind of structures [102], remarking again the importance of the different timescales in this kind of phenomena. An asymmetric social configuration allows the function to be optimized with ethnic markers, and the arena where cultures may compete, reinforcing the roles of their individuals, and building therefore, the different social profiles. With respect to the experimental part, the results that we obtain suggest that ethnic markers experiment should be replicated in contexts with different social configurations, studying, again, as the title suggests, the dependence on the context. It would be interesting the replication of simple ethnic separation altogether with a hierarchy or some reputation system. Once more, following the spirit of all the thesis, connects the importance of the ethnic, personal identifications with the social organization, that regulates its importance in the social equilibrium. In view of these conclusions, future works in this field should go deeper into the importance of the geometric social profile of the ego networks as an individual identifier. In this case, more data needs to be collected, in order to research other sociological variables like different aspects of culture, religion, political views or socioeconomic measures. Note that in the case of the present work, we have focused on the nationality, as the dataset provided us with a diverse sample of nationalities and the control group. In order to study other sociological variables, like wealth, race, gender or a particular social norm, new data is needed that fits those necessities. Moreover, we have focused on networks centered in the ego, where we have a perspective on the close social structure. It would be also interesting to study how much information is needed in order to study different kinds of networks. This is interesting from a practical point of view, as it could help to use different datasets to provide further proofs of the existence of the social signature, but also to check more properties of this measure. It would be interesting, for example, to study if family networks provide us enough information to talk about the different social values and/or material conditions of different families. If the information about the signatures still can be extracted, new datasets could be reused to make a stronger point about the cultural signature, saving the time of collecting new ego networks. Another possible path to follow in the future could be the study of signed networks together with the cultural profile. In our works, we have studied the importance of the social signature in order to determine the cultural profile, and its importance in the outcome of a relationship. It would be interesting to show the connection between these two elements, providing a relationship between cultural or socioeconomic variables and the friends or enemies an individual may have. The final suggestion for future work is related to the measure of properties of this social signature.We have stressed in the introduction (and in several chapters of the thesis) that the cultural values are a holistic product, which may be assumed and created by collectivities. In this work we have focused on a model that is based in the relationship between individual and its environment (in a mean-field like approach), but this is not needed. Considering again the example of the family, perhaps the family could be understood as a whole, and study the problem of its close social environment from a perspective of networks. Given that the problem of culture exists in several scales, it would be interesting to check if new relationships arise from this point of view. At the end of the day, all of these suggestions are basically related to checking the potential of the measure. This poses also an ethical question: if the potential of the social signature is shown in more contexts, and it correlates with personal information, it is important that these studies (public or private) are carried with the utmost concert about their possible implications and misuses. Programa de Doctorado en Ingeniería Matemática por la Universidad Carlos III de Madrid Presidente: Jesús Gómez Gardeñes.- Secretario: Alberto Antonioni.- Vocal: Andreia Sofía Monteiro Teixeira

Mención Internacional en el título de doctor This doctoral thesis was supported through the FPI contract BES-2017-079755 associated to the grant FIS2016-78313-P from Ministerio de Ciencia e Innovación (MCIN)/Agencia Estatal de Investigación (AEI)/ 10.13039/501100011033/ and FEDER Una manera de hacer Europa. Additionally the author also acknowledge finantial support from MCIN/AEI/10.13039/501100011033 through grant BADS, no. PID2019-109320GB-100 Programa de Doctorado en Ingeniería Matemática por la Universidad Carlos III de Madrid Presidente: Luis Guillermo Morelli.- Secretaria: María Pilar Guerrero Contreras.- Vocal: Vicente Mariscal Romero

Programa de Doctorado en Ingeniería Matemática por la Universidad Carlos III de Madrid Presidente: José Navarro Salas.- Secretario: Salvador José Robles Pérez.- Vocal: Iñaki Garay Elizondo

Mención Internacional en el título de doctor Online web services have grown dramatically in size and diversity in the last years, becoming essential components of our daily life and allowing us to conduct elementary tasks like working, getting informed, or keeping in contact with relatives and friends. However, all the changes and evolution experimented on by the online web services had not have been possible without implementing a profitable economic model that sustains it. Despite a suitable percentage of these services being fee-based, they represent a lucrative business that generates billions of dollars, allowing the creation of some of the biggest companies in the world in terms of market capitalization, like Alphabet Inc. or Meta Inc. (Previously known as Facebook Inc.). Being costless and lucrative is possible due to an advertising-based monetization model, which consists of delivering ads to the users in exchange for their services (e.g., Facebook or YouTube). Although online advertising dates back to the middle of the 90s, its popularity has experienced an increase among brands and advertising agencies in the last decade, mainly due to its capacity to reach precise audiences at a low cost. Converting online web services into advertising walls is a double-edged sword for the users. The capacity offered by online advertising to segment their audiences requires a massive collection of personal data from the users, including their web browsing histories or even more invasive data such as age, gender, or location to infer the online profile of the users. This data collection is possible due to implementing a complex tracking ecosystem by online advertising companies from which multiple stakeholders collect, process, and exchange information. The many privacy cases of abuse inflicted by this industry motivated the implementation of new regulatory efforts to protect consumers’ privacy in the last years. Some notable examples are the General Data Protection Regulation (GDPR) [1] in the European Union or the California Consumer Privacy Act Regulations (CCPA) [2] in California, USA. Further, these privacy regulations typically contain specific provisions and strict requirements for websites that provide sensitive material to end users, including sexual, religious, and health services. Implementing new regulatory frameworks, alongside the growth of online web services, forces an endless evolution of current techniques to study and audit online web services. Furthermore, there is a need to emphasize the online advertising ecosystem, as it represents the primary economic support of a high percentage of web services. Also, the activities and abuses conducted by this ecosystem drove the implementation of current privacy regulations to control the use and collection of personal data. This dissertation falls within the topics of Internet measurements, tackling the need for new measurement techniques and methodological approaches to audit and study online web services. These efforts want to increase the limited knowledge about web subsystems offering sensitive material, including their regulatory compliance regarding current privacy regulations. Also, this dissertation tackles the need to study and measure how big ad tech companies create and use the online profiles of their users to distribute tailored ads. Furthermore, the work presented in this dissertation raises the need for a more in-depth understanding of fundamental tools for conducting Internet measurement works, including their limitations and suitability for academic research. Specifically, this dissertation presents three main contributions: The first one corresponds with implementing a novel methodology to audit sensitive web services’ privacy, transparency, and regulatory compliance. We validate our method by looking at pornographic websites concerning the GDPR in the European Union. We focus our analysis on such types of websites for two main reasons: (i) the GDPR establishes specific provisions and strict requirements on sensitive websites, including pornographic ones. (ii) big ad tech companies set strict constraints for porn-related publishers. As a result, it opened new market opportunities for other actors who have specialized in advertising and tracking technologies for adult sites, creating a semi-decoupled ecosystem from the rest of the web. We perform a holistic analysis of over 6,843 pornographic websites, finding a prevalent absence of regulatory compliance and very extended use of tracking techniques, including advanced ones such as fingerprinting. These results stress the importance of studying the World Wide Web subsets that have not been scrutinized by regulators, policymakers, and the research community in depth. Second, we empirically and comprehensively analyze 13 domain classification services to study their labeling strategy and performance. These services have multiple applications, from business applications such as online advertising to academic research works to conduct category-dependent measurements or to identify the purpose of a website or online service. We study each domain classification service’s methodologies, scalability limitations, label constellations, and suitability for academic research studies. In some cases, their findings depend on the results provided by the domain classification services. We find that the limitations and shortcomings of each domain classification service heavily affect their suitability and applicability, both for practical solutions and academic studies. In the third and last contribution, we implement a novel methodology with real users to study the performance and quality of the profiling and ad targeting algorithms from the two most important stakeholders in the online advertising business, Google and Meta (previously Facebook). We find that half of the categories associated with the profiles are incorrectly assigned. We also observe the presence of sensitive categories in Facebook users, posing a privacy risk and potential regulatory noncompliance. In summary, this dissertation brings new methodologies and results to increase our limited knowledge about the web. This work has been supported by IMDEA Networks Institute. Programa de Doctorado en Ingeniería Telemática por la Universidad Carlos III de Madrid Presidente: Nikolaos Laoutaris.- Secretario: Hamed Haddadi.- Vocal: Leyla Bilge

Mención Internacional en el título de doctor Recent technological advances have led to an exponential growth in the volume of data generated. The quest to make sense of these data, some of which are usually complex, has led to recent interest in development of statistical methods for analysing data with complex structures. One such field of interest is functional data analysis (FDA), which deals with the analysis of data that can be considered as functions, curves, or surfaces observed over a domain set. Outlier detection is a challenging but important part of the exploratory analysis process in FDA because functional observations can exhibit outlyingness in various ways compared to the bulk of the data. This thesis addresses the problem of detecting and classifying outliers in functional data with three main contributions. First, the fdaoutlier R package is presented in Chapter 2. The package contains implementations of some of the state-of-the-art functional outlier detection methods in the literature. Some of the methods implemented include directional outlyingness, magnitude-shape plot, sequential transformations, total variation depth, and modified shape similarity index. Detailed illustrations of the functions of the package are provided, using various simulated and real functional datasets curated from the functional outlier detection literature. Overviews of the functional outlier detection methods implemented in the package are also presented in Chapter 2. This chapter therefore, serves as a review of some of the current literature in outlier detection for functional data. Next, two new methods, named ‘Semifast- MUOD’ and ‘Fast-MUOD’, are presented in Chapter 3. These methods work by computing for each curve three indices (magnitude, amplitude and shape index) that measure the outlyingness of that curve in terms of its magnitude, amplitude and shape. ‘Semifast- MUOD’ computes these indices with respect to (w.r.t.) a random sample of the dataset, while ‘Fast-MUOD’ computes these indices w.r.t. to the point-wise or L1 median. The classical boxplot is then used as a cutoff on the three indices to identify curves that are outliers of different types. A byproduct of the methods is an unsupervised classification of the outliers into different types, without the need for visualisation. Performance evaluation of the methods, using various real and simulated datasets, shows that Fast-MUOD is the better of the two new proposed methods for outlier detection, in addition to being very scalable. Comparisons with latest functional outlier detection methods in the literature also show superior or comparable outlier detection performance. In Chapter 4, some theoretical properties of the Fast-MUOD indices are presented. These include some definitions of the indices, as well as convergence proofs of the sample approximations. Some properties of the indices under simple transformations are also presented in this chapter. Finally, three techniques are presented in Chapter 5 for extending the Fast-MUOD indices to outlier detection in multivariate functional data observed on the same domain. These techniques include the use of random projections and identifying outliers on the marginal components of the multivariate functional data. The use of random projections showed the best result in performance evaluations with various real and simulated datasets. Chapter 6 contains some concluding remarks and possible future research work. This work has been supported by IMDEA Networks Institute Programa de Doctorado en Ingeniería Matemática por la Universidad Carlos III de Madrid Presidente: Francisco Javier Prieto Fernández.- Secretario: Alba María Franco Pereira.- Vocal: Fabian Scheipl

Wireless communication is a transformative technology that has changed the way we work, communicate and enjoy our free time. The number of connected devices is expected to increase to 29.4 billion by 2030. As a result, there is a demand for higher data rates, lower delays and constant connectivity of wireless devices. These demands have driven the networking community to seek new technologies as these requirements are beyond the capabilities of current networks. Communication at higher frequencies, beyond 10 GHz where most of current communications systems operate, could be the game changing technology for the next generation of networks. At millimeter-wave frequencies, 30-300 GHz, the available spectrum is larger than all spectrum currently allocated to cellular and wireless area networks (WLAN). The unlicensed spectrum at 60 GHz alone can offer 10 to 100 times more spectrum than it is available in current unlicensed WLANs. The larger bandwidth allocations allow for increased datarates. These multi-gigabit per second rates and milisecond latencies are now achievable thanks to new directional high gain antennas and cost-effective CMOS technology that can operate at mm-wave frequencies. However, with the use of this new technology new challenges arise. For example, since the communication is directional in mm-wave, the best direction to communicate needs to be decided, and updated in mobility scenarios. Aside from the technical deployment challenges at this frequency, another challenge is how to provide increased capabilities for the network such as localization and passive localization. The specific characteristic of propagation at mm-wave frequencies such as the quasi optical patterns, the dense deployments and the multi path diversity warrant designs of localization schemes aimed at exploiting these characteristic. Finally, stepping away from conventional network design, the future generation of networks can be enhanced by Machine Learning. Machine Learning has shown to be a powerful tool in many domains and has increased the autonomy of chatbots and self-driving cars. As we go further in frequency, ML can help design the different layers of the future communication networks to embrace the hardware imperfections, that are more relevant at higher frequencies, and to serve applications in the optimum way in different environments. The thesis is divided into two parts, first we study the challenges in the optimization and features of conventional mm-wave networks, and second, we consider a new way of designing mm-wave communications using Machine Learning. In the first part, we study the challenge of scaling mm-wave networks to dense deployments, and secondly we increase the capabilities of mm-wave networks by proposing a passive localization mechanism that is standard compliant. Currently the beam training mechanisms deployed to establish a directional link result in a large overhead in communications and this prohibits dense deployments. This is specially due to station beam training. While the APs can beam train with multiple stations simultaneously, the stations have to train one at a time. This overhead is even larger in mobile deployments where the devices require constant re-training to maintain communication. To overcome this overhead and allow dense deployments we propose a low-overhead beam training mechanism, SPIDER, that eliminates the station beam training overhead entirely. To this end, stations carry out phase-coherent measurements by switching through multiple receive beam patterns on a time-scale of tens of nanoseconds when receiving a packet preamble to calculate the AoA to the AP and select the best beam pattern for communication. Our results show that our algorithm achieves highly accurate angle estimation used to drive the beam steering decisions and that it reduces the overhead by an order of magnitude compared to IEEE 802.11ad beam training. The second work presented in the first part of this thesis is regarding the use of the network as a sensor in mm-wave frequencies. Joint communication and sensing approaches would allow for an extremely distributed cost-effective sensor network. At higher frequencies, due to the larger bandwidth, the theoretical precision of localization information is higher than at lower frequencies. However, localization algorithms for mm-wave frequencies can not directly build from the lower frequency ones. At lower frequencies, these algorithms leverage information from multiple antennas with separated RF chains, while in mm-wave phased antenna arrays are usually connected to a single RF chain, so the phase information per antenna can not be obtained. However, several properties of mm-wave communications have implications in the design of localization systems. The high path loss attenuation due to the high frequency implies that the line of sight paths (LOS) can be distinguished from the NLOS paths. Additionally, reflections on surfaces have limited scattering. These properties generate channels with sparse multi paths, for which geometrical propagation assumptions become appropriate. Additionally, the need for directional communication due to the high path loss results in the need for mechanisms that enable beam selection, which are the focus of the earlier works. Such beam training mechanisms become relevant for localization in mm-wave frequencies. Since during the beam training the environment is scanned using beam patterns in different directions, these information can be used to detect and track moving reflections. We present POLAR, the first passive mm-wave localization algorithm for mm-wave commercial devices with zero overhead. We use commercial APs whereas the station design is based on a full-bandwidth 802.11ad compatible FPGA-based platform with a phased antenna array. The stations exploit the preamble of the beam training packets of the APs to obtain channel state information measurements for all antenna patterns. With this, we determine distance and angle information for the different multipath components in the environment to passively localize a mobile object. We evaluate our system with multiple APs and a moving robot with a metallic surface. Our results show that is is possible to obtain 6.5cm mean error accuracy and sub-meter accuracy in 100% of the cases performing joint localization and sensing in a mm-wave testbed. In the second part of this thesis, we study a new way to design communications with the concept of AI-native air-interface framework in mm-wave. In this area of research, some works have proposed to learn to communicate using an end-to-end communication approach using deep learning (DL). However, a channel model is needed in order to train these models, and analytical channels were shown to produce a reduced performance in experimental scenarios. In order to extend this to real channels, model-free approaches have been presented in which the model can be learned implicitly from data. Building on this works to obtain an explicit channel model would also allow for transfer learning to speed up the training on time-variant channels. In this thesis, we present a neural network architecture based on Mixture Density Networks to explicitly learn the channel model for end-to-end communications for SISO and MIMO systems. Our results show that it is possible to learn complex stochastic channel models from data with differentiable methods that are compatible with the training of the autoencoder structure. These end-toend learning methods will become increasingly important in future generations of wireless networks, where the system complexity will increase due to a higher number of antennas, higher frequencies, wider bandwidths, and more complex hardware, making ML-based physical layer design highly appealing. Second, we study the scaling properties of end-to-end ML models for large-scale MIMO systems. We conclude that the bottleneck that prevents the scaling of such networks, in the state-of-the-art architectures, is due to the input and output representations. Large MIMO systems constitute highly demanding scenarios in which training would require unmanageable memory resources. We study typical approaches available in DL to reduce the memory allocation and show that a bit-wise architecture for MIMO allows to reduce the memory allocation by several orders of magnitude. Our results show that with the bit-wise architecture it is possible to extend the end-to-end approaches to large systems of antennas. This study is an important first step towards developing practical end-to-end learning algorithms for large scale MIMO systems. So far, end-to-end systems have only been applied in a per symbol manner as training these systems for blocks of symbols is complex as it introduces implicit channel coding. Optimizing the receiver only would allow to reduce the complexity and train in blocks of symbols while helping to deal with imperfections and other non-linearities without having to explicitly correct them. We explore the gains that can be achieved by using an ML based design on the receiver of single carrier mm-wave systems for frequency selective channels, where due to the multi-path and the resulting inter symbol interference block wise treatment is required. Our results show that it is possible to outperform baseline typical receivers in static scenarios. In summary, mm-wave technologies will open the door to the next generation of applications, however, new challenges need to be tackled. We provide new solutions to enable scalable deployments and show that the beam training station overhead can be completely eliminated. We introduce the feature of passive localization and sensing for mm-wave frequencies, and show the high accuracy achievable in an experimental FPGA based testbed. Finally, explore new ways of designing the physical layer to reduce the effect of hardware imperfections in mm-wave and show gains compared to state-of-the-art receivers. This work has been supported by IMDEA Networks Institute Programa de Doctorado en Ingeniería Telemática por la Universidad Carlos III de Madrid Presidente: Tobias Mirco Koch.- Secretario: Andrés García Saavedra.- Vocal: Dani Korpi

This thesis investigates the metric and tetrad formulations of three gravitational field theories in manifolds with timelike boundaries within the covariant phase space program. With the recently developed relative bicomplex framework, we explore the space of solutions and presymplectic structures associated with each action principle and analyse their equivalence. The first action we consider is the Einstein-Hilbert (EH) action with the Gibbons- Hawking-York boundary term. By including the appropriate boundary terms in the variational principles, we show that the metric and tetrad formulations derived from them are equivalent. Furthermore, we show that their solution spaces are the same and that their presymplectic structures and associated charges coincide. The second action we consider is the Palatini action with the Obukhov boundary term, assuming torsion and non-metricity, and we prove the equivalence between its metric and tetrad formulations. Furthermore, we show that the metric and tetradsector of the first-order Palatini formulation are equivalent to the metric and tetrad formulations of the EH action. Lastly, we introduce the Hojman-Mukku-Sayed (HMS) action, a generalisation of the Palatini action plus the Holst term in the presence of boundaries with non-metricity and torsion. We prove that the space of solutions of the HMS and Palatini actions coincided and conclude that HMS’s metric and tetrad sectors are identical to their corresponding versions of the EH action. Additionally, we prove that the Palatini and HMS Lagrangians are not cohomologically equal despite defining the same space of solutions. Consequently, a careful analysis is required for the presymplectic structures and the charges because they may differ. However, we show that the covariant phase spaces of both theories were equivalent. This sheds light on some open problems regarding the equivalence of their associated charges in different formulations. Programa de Doctorado en Ingeniería Matemática por la Universidad Carlos III de Madrid Presidente: José Navarro Salas.- Secretario: Bogar Díaz Jiménez.- Vocal: Mercedes Martín Benito

Mención Internacional en el título de doctor This research has been funded in part by MCIN/AEI/10.13039/501100011033 and by “ERDF A way of making Europe” through grant PGC2018-098186-B-I00 (BASIC). Programa de Doctorado en Ingeniería Matemática por la Universidad Carlos III de Madrid Presidente: Sandro Meloni.- Secretaria: Francesca Lipari.- Vocal: Giulia Andrighetto

Mención Internacional en el título de doctor El principal objetivo de esta tesis es presentar y probar la viabilidad de un método no estándar para controlar el estado de un sistema cuántico, cuya dinámica está gobernada por la ecuación de Schrödinger, modificando sus condiciones de frontera en lugar de interaccionar con el sistema a través de campos externos para dirigir su estado. El control del estado de sistemas cuánticos está volviéndose más y más significativo dado el asombroso avance experimental que está teniendo lugar motivado por la carrera para alcanzar tecnologías cuánticas efectivas. Un requisito natural para cualquier procesador de información cuántico es el control de los estados de un cúbit. Por ejemplo, los cúbits basados en espín pueden controlarse a través de puertas universales basadas en la rotación del espín un cierto ángulo sobre un eje dado [60]. Una demostración de la viabilidad del control y de la lectura de alta fidelidad de un cúbit basado en espín puede encontrarse en [55, 59]. El marco matemático necesario para trabajar con el control de sistemas cuánticos basados en el espín es la Teoría de Control Geométrica (véase [19] para un repaso de este enfoque). También puede mencionarse [15, 56], dónde se realiza un estudio del control de sistemas cuánticos de espín utilizando dicha teoría. Una revisión del estado de la cuestión en control (óptimo) de sistemas cuánticos puede verse en [31]. Sin embargo, la Teoría de Control Geométrica encuentra problemas serios a la hora de tratar sistemas infinito dimensionales provenientes de las dificultades intrínsecas a la geometría en dimensión infinita. Éste constituye un problema estándar que se aborda también, por ejemplo, en Teoría de Campos y puede por lo tanto ser sorteado. Sin embargo, lo cierto es que tan sólo un pequeño número de resultados han aparecido en las fuentes bibliográficas, sobre todo referentes a la controlabilidad de sistemas bilineales (cf. sec. 3.2). Véanse por ejemplo los trabajos de Beauchard et al. [11, 12] y Chambrion et al. [18]. El enfoque habitual del control de sistemas cuánticos se basa en el uso de campos externos para manipular el estado del sistema. Desde un punto de vista tecnológico, aparecen dificultades al intentar controlar un sistema cuántico de este modo que tienen que ver con las complicaciones para manipular un sistema hecho de unas pocas partículas mientras se intenta mantener las correlaciones cuánticas. Como consecuencia, el sistema cuántico debe mantenerse a muy baja temperatura y las interacciones debe producirse muy rápidamente [5], lo cual es inconveniente para las aplicaciones. El paradigma de Control Cuántico en la Frontera aborda el control de sistemas cuánticos de una forma diametralmente opuesta al enfoque estándar. En lugar de buscar el control del sistema cuántico interactuando directamente con el sistema a través de un campo externo, el control se consigue manipulando las condiciones de contorno del sistema. El espectro de un sistema cuántico, por ejemplo un electrón moviéndose en un pozo de potencial, depende de las condiciones de contorno (típicamente Dirichlet o Neumann). Por lo tanto, modificando dichas condiciones de contorno puede modificarse el estado del sistema, lo cual permitiría en última instancia controlarlo [34]. Este tipo de interacción, que es más débil en cierto sentido, hace esperar que sea más sencillo mantener las correlaciones cuánticas. Llamamos sistema (cuántico) de control en la frontera a los sistemas definidos dentro de este paradigma de control. El marco establecido por el Control Cuántico en la Frontera ha sido usado para mostrar cómo generar estados entrelazados en sistemas compuestos modificando las condiciones de frontera del sistema [38]. La relación entre dicho paradigma y la topología del sistema ha sido explorada en [58] y recientemente usada para describir las propiedades físicas de sistemas con paredes móviles [25–28,30]. Sin embargo, a pesar del interés intrínseco que tiene, problemas básicos como la controlabilidad de sistemas sencillos dentro de este paradigma no ha sido aún estudiada. La mayoría de los sistemas cuánticos que aparecen en las aplicaciones para desarrollar ordenadores cuánticos, tales como las trampas de iones o los circuitos superconductores, son infinito dimensionales. Mientras tanto, los modelos usados para describirlos son aproximaciones finito dimensionales [49, 70], lo que introduce errores en dicha descripción. Es por tanto natural buscar mejores modelos matemáticos para estos sistemas. Buenos candidatos para estos modelos pueden construirse a partir de lo que llamamos Circuitos Cuánticos, una generalización de los grafos cuánticos (esto es, grafos métricos equipados con un operador diferencial en sus aristas junto con condiciones de contornos apropiadas que definan una extensión autoadjunta de dicho operador [46,47]). Por un lado, los Circuitos Cuánticos comparten con los dispositivos físicos la estructura tipo grafo y la dinámica en ellos está gobernada por operadores diferenciales (en concreto, consideraremos el operador de Laplace-Beltrami y Laplacianos magnéticos junto con algunos potenciales que representan campos magnéticos y eléctricos en los Circuitos Cuánticos). Por otro lado, los Circuitos Cuánticos son suficientemente sencillos como para encontrar soluciones de la ecuación de evolución numéricamente (o incluso analíticamente). Todo esto hace interesante considerar la dinámica y la controlabilidad de sistemas definidos en Circuitos Cuánticos. Algunos ejemplos concretos de Circuitos Cuánticos pueden encontrarse al final del Capítulo 6, véase la Figura 6.1. Más aún, los grafos cuánticos pueden verse como un caso concreto de Circuito Cuántico, por lo que los resultados presentados en esta tesis pueden aplicarse directamente a ellos. Y, además de el interés matemático de estudiar la dinámica de sistemas definidos en grafos cuánticos, creemos que éstos pueden ser utilizados para modelizar circuitos superconductores. Más concretamente, es sabido que las junturas de Josephson (que son un dispositivo fundamental en los circuitos superconductores, cf. [22, 70, 71]) pueden modelizarse usando condiciones de frontera [48,68]. Nótese que los circuitos superconductores son una de las prometedoras nuevas tecnologías en el desarrollo de ordenadores cuánticos. Para mostrar la viabilidad del método de Control Cuántico en la Frontera, definimos una familia de sistemas de control cuántico en la frontera sobre Circuitos Cuánticos. Para ello, utilizamos la caracterización de las extensiones autoadjuntas del operador de Laplace-Beltrami desarrolladas en [37], que parametriza las extensiones autoadjuntas en función de operadores unitarios actuando en el espacio de datos en la frontera. Estos operadores unitarios son adecuados para representar los grupos de simetría de la variedad subyacente [36], hecho que fue usado en [8] para caracterizar las condiciones de frontera compatibles con las estructuras de grafo que consideramos aquí. Antes de poder abordar el problema de controlabilidad, es necesario afrontar el problema de existencia de soluciones de la ecuación de Schrödinger con condiciones de contorno dependientes del tiempo, problema que tiene su propia importancia. Situaciones similares a la que presentaremos han sido consideradas anteriormente para algunos sistemas cuánticos concretos [21, 23, 58]. La dependencia temporal del dominio del Hamiltoniano del sistema añade dificultades extra a las habituales complicaciones que surgen de la naturaleza discontinua de los operadores no acotados, lo que compromete la existencia de soluciones para la ecuación de Schrödinger. En este sentido, los resultados más generales de que disponemos son debidos a J. Kisyński [44]. Basándose en una idea aproximativa de K. Yosida [72], Kisyński propone condiciones suficientes para la existencia de operadores de green para problemas de Cauchy abstractos en espacios de Banach, incluyendo el caso en el que los operadores que definen la evolución tienen dominios dependientes del tiempo. Estos resultados pueden ser aplicados al caso concreto de la ecuación de Schrödinger, tal y como hace el propio Kisyński, cf. [44, Sec. 8]. Más concretamente, proporciona condiciones suficientes para la existencia de propagadores unitarios para sistemas cuánticos cuyos Hamiltonianos tienen dominio dependiente del tiempo pero no así las formas sesquilineales asociadas a éstos. Este caso también fue estudiado por B. Simon [65], cuyo enfoque se basa en el mismo método aproximativo de Yosida, pero abordado el problema de manera ligeramente distinta. Para simplificar la exposición, en lugar de aplicar los resultados más generales de Kisyński directamente a nuestros sistemas cuánticos de control en la frontera, introduciremos una familia de sistemas de control estándar relacionada que llamamos sistemas de inducción cuántica, cuyos Hamiltonianos tendrán dominio de forma constante. La relación entre estas dos familias de sistemas es usada frecuentemente en la discusión, lo que nos permite probar resultados primero para los sistemas de inducción cuántico y transferirlos a los sistemas de control en la frontera. Siguiendo esta estrategia, usaremos los resultados de Simon y de Kisyński para Hamiltonianos con dominio de forma constante para proporcionar condiciones suficientes para que ambos sistemas tengan dinámica bien definida. En el estudio de las propiedades dinámicas de los Hamiltonianos dependientes del tiempo con dominio de forma constante, la escala de espacios de Hilbert asociada a el Hamiltoniano, H+ ⊂ H ⊂ H−, aparece como una herramienta básica tanto en el desarrollo de Simon como en el de Kisyński. A través de esta tesis, usamos nociones como la existencia de dinámica o la controlabilidad tanto en H como en H−; por ello, usaremos los adjetivos fuerte y débil para referirnos a las nociones en H y en H− respectivamente. Una vez resuelto el problema de la existencia de soluciones de la ecuación de Schrödinger, podremos abordar el problema de controlabilidad. Usaremos como una herramienta útil para este fin el Teorema 4.3.4, un resultado de estabilidad que generaliza los resultados de A.D. Sloan [66] constituyendo uno de los resultados originales principales expuestos e esta tesis. Que sepamos, esta generalización es el resultado de estabilidad más general en el contexto de la ecuación de Schrödinger dependiente del tiempo. La demostración de este teorema se basa en las técnicas usadas por Simon y, en la prueba de dicho resultado, obtenemos una ligera generalización de algunos resultados en [65] (cf. Lemma 4.3.7). La importancia de este resultado de estabilidad no reside tan solo en su papel a la hora de demostrar la controlabilidad de los sistemas de control bajo estudio permitiendo demostrar la viabilidad del Control Cuántico en la Frontera; tiene consecuencias de mayor alcance para la Teoría de Control Cuántico ya que permite obtener cotas a priori del error cometido al controlar el estado del sistema. Demostramos la controlabilidad aproximada de los sistemas de inducción cuántica basándonos en un resultado de T. Chambrion et al. in [18], donde estudian la controlabilidad de una clase de sistemas de control cuántico bilineales con controles constates a trozos. La estructura que tienen los sistemas de inducción cuántica es similar, pero no exactamente igual, a la de los problemas estudiados por Chambrion et al. Por ello, definimos un sistema auxiliar para el cual el teorema de Chambrion et al. asegura controlabilidad aproximada y después, usando el resultado de estabilidad, mostramos que esta controlabilidad aproximada se extiende al sistema de inducción cuántica original. Si embargo, la clase de controles considerada por Chambrion et al. no puede ser llevada automáticamente a sistemas con Hamiltonianos cuyos dominios dependen del tiempo, ya que las soluciones de la ecuación de Schrödinger (en el sentido fuerte) podrían no existir. Por lo tanto, la controlabilidad aproximada obtenida para los sistemas de inducción cuántica usando este procedimiento es en el sentido débil (es decir, en H−). Usando de nuevo el resultado de estabilidad podemos demostrar controlabilidad aproximada en sentido fuerte para los sistemas de inducción cuántica con controles suaves, y ésta puede ser transferida a los sistemas de control cuántico en la frontera usando la relación entre ambos. Estos resultados de controlabilidad, demostrados en el Capítulo 7, junto con los resultados de estabilidad en el Capítulo 4 constituye las contribuciones originales más relevantes de esta tesis. Por último, remarcamos el hecho de que la caracterización de las extensiones autoadjuntas que usamos aquí puede ser aplicada a otros operadores diferenciales como el operador de Dirac [35, 40, 57], por lo que los resultados presentados en esta tesis podrían extenderse a tales casos. Más aún, dicha caractrerización está adaptada al cálculo numérico [39, 51], por lo que podrían desarrollarse algoritmos con aplicaciones al control óptimo de éstos sistemas a partir de los resultados aquí mostrados. This work was partially supported by the “Ministerio de Economía, Industria y Competitividad” research project MTM2017-84098-P, the QUITEMAD project P2018/TCS-4342 funded by “Comunidad Autónoma de Madrid” and by “Universidad Carlos III de Madrid” through its Ph.D. program grant PIPF UC3M 01-1819 and through its mobility grant in 2020. Programa de Doctorado en Ingeniería Matemática por la Universidad Carlos III de Madrid. Presidente: Fernando Lledó Macau.- Secretario: Paolo Facchi.- Vocal: Thomas Chambrion

Topological indices have been widely used in different fields associated with scientific research. They are recognized as useful tools in applied research in Chemistry, Ecology, Biology, Physics, among others. For many years, scientists have been trying to improve the predictive power of the famous Randi’c index. This led to the introduction and study of new topological descriptors that correlate or improve the level of prediction of the Randi’c index. Among the most commonly used descriptors are the Inverse index, the first general Zagreb index and the recently introduced Arithmetic- Geometric index. In this work we study the mathematical properties and relationships of the aforementioned topological indices. Programa de Doctorado en Ingeniería Matemática por la Universidad Carlos III de Madrid Presidente: Domingo de Guzmán Pestana Galván.- Secretaria: Ana Portilla Ferreira.- Vocal: Eva Tourís Lojo

This thesis is mainly motivated by the analysis, understanding, and prediction of human behaviour by means of the study of their digital fingeprints. Unlike a classical PhD thesis, where you choose a topic and go further on a deep analysis on a research topic, we carried out a breadth analysis on the research topic of complex networks, such as those that humans create themselves with their relationships and interactions. These kinds of digital communities where humans interact and create relationships are commonly called Online Social Networks. Then, (i) we have collected their interactions, as text messages they share among each other, in order to analyze the sentiment and topic of such messages. We have basically applied the state-of-the-art techniques for Natural Language Processing, widely developed and tested on English texts, in a collection of Spanish Tweets and we compare the results. Next, (ii) we focused on Topic Detection, creating our own classifier and applying it to the former Tweets dataset. The breakthroughs are two: our classifier relies on text-graphs from the input text and we achieved a figure of 70% accuracy, outperforming previous results. After that, (iii) we moved to analyze the network structure (or topology) and their data values to detect outliers. We hypothesize that in social networks there is a large mass of users that behaves similarly, while a reduced set of them behave in a different way. However, specially among this last group, we try to separate those with high activity, or low activity, or any other paramater/feature that make them belong to different kind of outliers. We aim to detect influential users in one of these outliers set. We propose a new unsupervised method, Massive Unsupervised Outlier Detection (MUOD), labeling the outliers detected os of shape, magnitude, amplitude or combination of those. We applied this method to a subset of roughly 400 million Google+ users, identifying and discriminating automatically sets of outlier users. Finally, (iv) we find interesting to address the monitorization of real complex networks. We created a framework to dynamically adapt the temporality of large-scale dynamic networks, reducing compute overhead by at least 76%, data volume by 60% and overall cloud costs by at least 54%, while always maintaining accuracy above 88%. Publicado Programa de Doctorado en Ingeniería Matemática por la Universidad Carlos III de Madrid Presidente: Rosa María Benito Zafrilla.- Secretario: Ángel Cuevas Rumín.- Vocal: José Ernesto Jiménez Merino

Mención Internacional en el título de doctor This thesis develops the discrete sub- and supersolutions method and applies it to prove the convergence of the nonlinear finite element method applied to the generalized di usive logistic equation...... Este trabajo ha sido desarrollado en el Departamento de Matemáticas de la Universidad Carlos III de Madrid (UC3M) bajo la dirección de la profesora Marcela Molina Meyer. Se contó con una beca de la UC3M de ayuda al estudio de máster de dos años de duración y posteriormente con un contrato de investigador en formación de la UC3M durante tres años más. Adicionalmente se recibió una ayuda de la UC3M para realizar una estancia de investigación, financiada por el programa "Ayudas para la movilidad de investigadores de la Universidad Carlos III de Madrid en centros de investigación nacionales o extranjeros, Convocatoria 2015. Modalidad B: investigadores en formación predoctoral". La estancia de investigación se realizó en el Departamento de Matematicas de Heriot-Watt University (Edimburgo). Además, esta tesis ha sido respaldada parcialmente por el Ministerio de Economía y Competitividad de España con el proyecto MTM2015-65899-P y por el Ministerio de Ciencia, Innovación y Universidades de España con el proyecto PGC2018-097104-B-100. Programa de Doctorado en Ingeniería Matemática por la Universidad Carlos III de Madrid Presidente: John Werner Peter Volker.- Secretario: Julia María Novo Martín.- Vocal: Sara Pollock

Mención Internacional en el título de doctor This Doctoral Thesis has been supported by the Severo-Ochoa grants SEV-2015- 0554 and BES-2016-077968, from the Spanish Ministry of Economy and Competition (MINECO), via the Instituto de Ciencias Matemáticas (ICMAT) of the Centro Superior de Investigaciones Científicas (CSIC). Additionally, the author has been partially supported by the Ministry of Economy and Competitiveness of Spain under Research Project MTM2017-84098-P. Programa de Doctorado en Ingeniería Matemática por la Universidad Carlos III de Madrid Presidente: Enrique Pardo Espino.- Secretario: Luis Alberto Ibort Latre.- Vocal: Nadia S. Larsen

Mención Internacional en el título de doctor In this thesis we present a pseudospectral method in an interval, in the disk and in the circular annulus. Unlike the methods already known, the disk is not duplicated. Moreover, we solve the Laplace equation under non-homogeneous Dirichlet, Neumann and Robin boundary conditions, by only using the elements of the corresponding differentiation matrices. It is worth mentioning that we do not use any quadrature, nor need to solve any decoupled system of ordinary di erential equations, nor use any pole condition, nor require any lifting. We also solve several numerical examples to show the spectral convergence. The pseudospectral method developed in this thesis is applied to estimate Sherwood numbers integrating the mass flux to the disk and we also implement it to solve Lotka-Volterra systems and nonlinear diffusion problems involving chemical reactions. Furthermore, we simulate numerically the classical positive solutions, large solutions and metasolutions of the degenerate diffusive logistic equation with spatial heterogeneities in an interval and in the disk through a collocation spectral method presented here. Some metasolutions of the different branches of metasolutions for the logistic equation introduced in [27] are simulated by the first time. Moreover, we simulate numerically through a collocation spectral method jointly with pathfollowing techniques the classical and non-classical non-negative branches of solutions of Lotka- Volterra competing systems allowing the presence of spatial heterogeneities. Some of these solutions are simulated for the first time. Also we prove the convergence of an innovative Chebyshev-Gauss-Lobatto( (CGL) pseudospectral method applied to fourth order boundary value problems. The proposed method enjoyed all the advantageous of the pseudospectral methods. Moreover, we can select (N - 3) interior CGL collocation points to enforce the equation, meanwhile we use the remaining four collocation points to assure the boundary conditions. In addition, the numerical methods introduced in this thesis are extremely innovative because they can be used to solve non-radially symmetric problems. The models are of a huge interest in Spatial Ecology because they enable us to analyse the effects of the spatial heterogeneity on the evolution of the terrestrial ecosystems. En esta tesis doctoral presentamos un método pseudoespectral en un intervalo, en el disco y en un anillo circular. A diferencia de otros métodos conocidos, el disco no está duplicado. Más aún, resolvemos la ecuación de Laplace bajo condiciones de frontera de tipo Dirichlet, Neumann y Robin, haciendo uso únicamente de los elementos de las matrices de diferenciación correspondientes. Vale la pena mencionar que no utilizamos ningún tipo de cuadratura ni necesitamos resolver ningún sistema desacoplado de ecuaciones diferenciales ordinarias, ni tampoco el uso de una condición de polo, ni siquiera se requiere un ”lifting”. También resolvemos numéricamente diversos ejemplos para mostrar la convergencia espectral. El método pseudoespectral desarrollado en esta tesis es aplicado para estimar el número de Sherwood integrando el flujo de masa en el borde del disco. También es implementado para resolver sistemas Lotka-Volterra de competición y problemas de difusión no lineales. Asimismo, simulamos numéricamente a través de un método espectral de colocación las soluciones clásicas positivas, las soluciones largas y las metasoluciones de la ecuación logística difusiva degenerada con heterogeneidades espaciales en un intervalo y en el disco. Algunas metasoluciones de diferentes ramas de metasoluciones para la ecuación logística introducidas en [27] son simuladas por primera vez. Además, simulamos numéricamente las soluciones clásicas no negativas y las soluciones no clásicas de los sistemas de competición de Lotka-Volterra a través de un método espectral de colocación junto con técnicas de continuación numérica de ramas de soluciones, permitiendo la presencia de heterogeneidades espaciales. Algunas de estas soluciones se simulan por primera vez. También probamos la convergencia de un método pseudoespectral innovador de Chebyshev-Gauss-Lobatto (CGL) aplicado a problemas de valores contorno de cuarto orden. El método propuesto disfruta de todas las ventajas de los métodos pseudoespectrales. Además, podemos seleccionar (N - 3) CGL puntos interiores de colocación donde forzar la ecuación, mientras utilizamos los cuatro puntos de colocación restantes para asegurar las condiciones de frontera. Los métodos numéricos presentados en esta tesis son extremadamente innovadores porque sirven para resolver problemas radialmente no simétricos. Además, los modelos son de un enorme interés en Ecología Espacial porque nos permiten analizar los efectos de las heterogeneidades espaciales en la evolución de los ecosistemas terrestres. Este trabajo ha sido desarrollado en el Departamento de Matemáticas de la Universidad Carlos III de Madrid (UC3M) bajo la dirección de la profesora Marcela Molina Meyer. Se contó con una beca de la UC3M de ayuda al estudio de máster y posteriormente con un contrato predoctoral de la UC3M. Adicionalmente se recibió una ayuda parcial del Ministerio de Ciencias, Tecnología y Universidades de España bajo proyecto de investigación MTM2015-65899-P. Programa de Doctorado en Ingeniería Matemática por la Universidad Carlos III de Madrid Presidente: J. Chris Eilbeck.- Secretario: Luis Francisco López Bonilla.- Vocal: Víctor Pereyra

Mención Internacional en el título de doctor Cancer kills 26.4% of Spanish people. It is the second cause of death, just behind diseases of the circulatory system, 28.3% [1]. The growth of new blood vessels from the existing vasculature in response to chemical signals from a tumor is called tumorinduced angiogenesis and it is closely related to cancer and metastasis. The growth rate of a tumor is considerably increased in its vascular stage compared to its avascular and solid stage, therefore treating cancer turns excessively difficult and the survival rates rapidly decrease [2]. Among diseases that cause disability but not substantial mortality, age-related macular degeneration may cause severe loss of vision or blindness in many people, particularly the elderly. It is projected that 196 million people will be affected by age-related macular degeneration in 2020, increasing to 288 million by 2040 [3], which is likely an underestimation [4]. With age, Bruch’s membrane gets thicker and some damaged cells in the retina become inflamed. The secretion of chemical signals from those cells due to their inflammation induces angiogenesis, but the new blood vessels are disorganized and leaky causing the loss of vision. John Hunter was the pioneer in describing the vessel formation process in 1787 [5], but the first person who coined the word “angiogenesis” was Arthur T. Hertig in 1935 [6]. He was studying the formation of new blood vessel in the primary placenta of the macaque monkey when this word was used for the first time. Years later, in 1971, Judah Folkman hypothesized that tumors emit Tumor Angiogenic Factors (TAF) to attract blood vessels to them [7]. This investigation triggered the research field of angiogenesis in cancer and in 1989 one of the most important angiogenic factors was discovered: the Vascular Endothelial Growth Factor (VEGF). Since then, drugs with antiangiogenic effects have been investigated for cancer, age-related macular degeneration and other diseases, as it is involved in more than seventy different diseases. However, angiogenesis also occurs in normal and vital processes such as wound healing or the growth of a fetus. The difference between physiological and pathological angiogenic processes is a matter of balance. In a healthy process, angiogenesis develops to its proper extent and then stops, while in pathological processes angiogenesis does not stop or it does not develop sufficiently. Angiogenesis keeps the number of blood vessels needed in balance: few blood vessels cause tissue death, while uncontrolled vascular proliferation can lead to cancer, macular degeneration and other diseases. Angiogenesis is a complex, multistep and well regulated process where biochemistry and physics are intertwined. The process entails signaling in vessel cells being driven by both chemical and mechanical mechanisms that result in vascular cell movement, deformation and proliferation. In a later stage of angiogenesis, vessel cells rearrange to form lumen and allow the perfusion of the blood inside the sprout. Depending on what induces the angiogenesis, different environments and cells should be considered, for instance in the retina. A detailed review of the processes involved in angiogenesis from the biological point of view is given in section 1.1. Beyond experimental investigations, mathematical models of angiogenesis try to help in understanding the process and how the relevant mechanisms of angiogenesis interact. The approach of some models focus on a single scale or a single process of those involved to deepen the knowledge about it. Others span multiple scales or the whole process to give an idea about how to prevent or favor angiogenesis. In section 1.2, we briefly review the mathematical models of angiogenesis that have been used to date as well as those when angiogenesis occurs in the retina and models of lumen formation, the late stage of angiogenesis. A crucial question about modeling is how to integrate the multiple scales and mechanisms present in angiogenesis in a mathematical model. A model is expected to be useful to explore methods for promoting and inhibiting angiogenesis. However, answering this question with this expectation is not a simple task. Assembling all the processes involved with their different time and length scales requires to develop a cellular dynamics model combined with models for the continuum fields. We achieved this objective by developing a hybrid cellular Potts model of early stage angiogenesis, given in chapter 2. In contrast to recent models, this mathematical and computational model is able to explore the role of biochemical signaling and tissue mechanics. A exhaustive description of the results of the numerical simulations complete the chapter 2. The advantages of discovering the reasons why angiogenesis starts in the retina or inhibitory mechanisms are innumerable. Unraveling the causes of neovascularization in the retina and giving possible solutions for age-related macular degeneration are our motivation to adapt the angiogenesis model of chapter 2 to the retina. In chapter 3, we present the model and the numerical results. If mathematical models of angiogenesis that incorporate multiple scales and cellular signaling processes are not that common, those that also include lumen formation are almost nonexistent. In chapter 4, we describe two models of lumen formation and their results. The lumen formation in the first model takes place in a already developed sprout. Although some restrictions in the model make its applications and possibilities limited, its study is convenient to establish the basis of the second proposed model. In this second model, the lumenization occurs while the sprout is developing and the pressure of the blood is involved, following recent experiments of lumen formation during angiogenesis. This model is work in progress, but we believe that showing the preliminary results in chapter 4 may be interesting. A critical step in the development of a mathematical and computational model is to analyze the viability of its simulations. The simulations of the model in chapter 2 have been carried out thanks to the parallel computing on Graphics Processing Units (GPUs), as well as simulations of chapters 3 and 4. The large amount of square elements of the grid, nodes, cells and sprouts make this type of computation suitable for these models. The way they have been implemented is explained in chapter 5. Finally, conclusions of this thesis and future work are drawn in the last chapter 6. This chapter highlights and summarizes the research that has been carried out and proposes future extensions and applications of this work. La investigación de esta tesis ha sido financiada por los proyectos de investigación del Ministerio de Economía y Competitividad (ahora FEDER/Ministerio de Ciencia, Innovación y Universidades–Agencia Estatal de Investigación) No. MTM2014-56948-C2-2-P y No. MTM2017-84446-C2-2-R. Programa de Doctorado en Ingeniería Matemática por la Universidad Carlos III de Madrid Presidente: Ana María Carpio Rodríguez.- Secretario: Filippo Terragni.- Vocal: Stephen W. Teitsworth

La presente tesis está dedicada al estudio de ciertos operadores de tipo integrodiferencial, de orden diferencial cercano a cero, de interés en aplicaciones y en teoría matemática.(...) El trabajo se divide en dos partes principales. En la primera parte estudiamos operadores lineales definidos por núcleos de Léevy no integrables poniendo el énfasis en núcleos con singularidad muy débil, en el límite de integrabilidad; en la segunda parte estudiamos operadores no lineales de tipo p-laplaciano fraccionario, incluyendo también núcleos débilmente no integrables. The present thesis is dedicated to the study of certain integro-differential operators with differential order close to zero, which are interesting both, from the theoretical point of view and for the applications.(...) This work is divided into two main parts. In the first part we study linear operators defined by non integrable Lévy kernels, placing the emphasis on kernels with a very weak singularity. In the second part we study nonlinear operators of the fractional p-Laplacian type, including also weakly non-integrable kernels. Programa de Doctorado en Ingeniería Matemática por la Universidad Carlos III de Madrid Presidente: Salvador Villegas Barranco.- Secretario: Cristina Brandle Cerqueira.- Vocal: Raúl Ferreira de Pablo

In chapter 2 we presented results obtained through experiments on a vibrated monolayer of granular metallic rods, focusing on the effects that introducing a circular obstacle to the centre of the cavity (thus creating an annulus) would have on the steady states of the system. Although topological arguments suggested that defects found on systems with no obstacle would not appear, we observed a separation of the system into regions consisting of smectic and tetratic ordering, the latter containing completely unordered point-like defects. We argue that the topological approach based on a continuum argument is rendered invalid by the size of the elements involved, in particular the ratio of the central obstacle size to particle lengths. The tetratic structures can be viewed as domain walls separating the smectic regions, which were invariably equal to four in our experiments. In chapter 3 we used the SPT to study two different systems of two-dimensional particles. In the first one, composed of particles shaped like hard isosceles triangles, we found, for the one component fluid, 1st and 2nd order Isotropic-Nematic transitions and for symmetric binary mixtures a 1st order Nematic-Nematic transition ending in a critical point and a Landau point where two first-order Nematic-Triatic transitions and a Nematic-Nematic demixing transition coalesce. We also found that for these particular mixtures a Triatic phase can be stabilized even though both species involved do no exhibit a stable Triatic phase on the one-component limit. To the best of our knowledge this is the first example of a Nematic-Nematic transition on one-component hard particle systems. For the second system, of hard rectangular particles, we found that polydispersity has an important effect on the phase diagram, strengthening the 1st order Isotropic-Nematic transition and decreasing the stability of the Tetratic phase. Also we found that polydispersity induces strong fractionation between the coexisting phases. Finally in chapter 4 we studied the effects of nano-confinement on a system of hardboard like particles. Following the restricted orientation (Zwanzig) approach we obtained a DF using the FMT to approximate the excess part of the free-energy of the system by means of numerical minimization. We considered a particular set of particle aspect ratios as well as slightly biaxial particles. We hoped to determine whether homeotropic configurations could be stabilized by the presence of two hard walls. We obtained that, for particular particle aspect ratios, homeotropic phases do stabilize and also that allowing for particle biaxiality greatly enriches the phase diagram. The results obtained can be explained by considering how entropy is maximized as the system approaches close packing. Programa de Doctorado en Ingeniería Matemática por la Universidad Carlos III de Madrid Presidente: Luis Mederos Martín.- Secretario: Daniel de las Heras Díaz-Plaza.- Vocal: Giorgo Cinacchi

Programa de Doctorado en Ingeniería Matemática por la Universidad Carlos III de Madrid Presidente: Francisco José Marcellán Español.- Secretario: Bernardo de la Calle Ysern.- Vocal: Manuel Bello Hernández

Mención Internacional en el título de doctor This thesis set out to contribute to the realm of social physics, with a particular focus on human social networks. Our approach, however, is somewhat di erent from what is typical in disciplines such as complex systems or statistical physics. Rather than simplifying the features of the constituents of our system (people), and stressing their rules of interaction, we focus on better understanding those very same constituents, modelling them as social atoms. Our rationale is that a better understanding of such an atom may shed light on how (and why) it interacts with other atoms to form social collectives. Given its robustness and the evolutionary roots of its premises, we use the Social Brain Hypothesis as our departure point. This theory states that the evolutionary drive behind the development of large brains in humans was the need to process social information and that the limited capacity of our brains imposes a limit to the number of relationships we can manage— the so-called “Dunbar’s number”, roughly 150. Moreover, evidence keeps revealing that these relationships are further organised in a series of hierarchically inclusive layers with decreasing emotional intensity, whose sizes exhibit a more or less constant scaling. Notwithstanding the empirical evidence, neither the presence of scaling in the organisation of personal networks nor its connection with limited cognitive skills had been explained so far. In Chapter 2 we present a mathematical model that solves this puzzle. The assumptions of the model are quite simple, and well founded on empirical evidence. Firstly, the number of relationships we maintain tends to be stable on average. Secondly, these relationships are costly, and our resources are limited. With these two premises, our results show that the hierarchical organisation emerges naturally from the principle of maximum entropy. Not only that, but we also predict a hitherto unnoticed regime of organisation whose existence we prove using several datasets from communities of immigrants. The former model considers that relationships can only belong to a discrete set of categories (layers). In Chapter 3 we extend it so that relationships are classified in a continuum. This modification allows us to test the model with data from very di erent sources such as online communications, face-to-face contacts, and phone calls. Our results show that the two regimes of organisation found in the previous model persist in this variant, and reveal the underlying existence of a (universal) scaling parameter which does not depend on any particular number of layers. To incorporate these ideas into socio-centric models, we build on the so-called Structural Balance Theory. This theory, underpinned by psychological motivations, posits that the structure of social networks of positive and negative relationships are highly interdependent. However, the theory has received little empirical validation, and negative social relationships are poorly understood—both from an ego-centric and a socio-centric perspective. For that reason, we turn to developing an experimental software in order to gather data within a school. In Chapters 4 and 5 we present results from these experiments. In Chapter 4 we analyse the socio-centric networks using machine learning techniques and find that the structure of positive and negative networks is indeed very much connected. Besides, we study the two types of networks separately, showing that they exhibit quite distinct features and that gender e ects in negative social networks are weak and asymmetrical for boys and girls. In Chapter 5, on the other hand, we focus on the structure of negative personal networks. Remarkably, using data from two di erent experimental settings, we show that the structure of personal networks of negative relationships mirrors that of the positive ones and exhibits a similar scaling—albeit their size is significantly smaller. Chapter 6 summarises our results and presents future (and current) lines of investigation. Among them, we outline a model of a social fluid that uses the insights gained with this thesis to build a model of social collectives as ensembles of personal networks. This model is compatible, at the micro-level, with the observations of the social brain hypothesis, and, at the macro-level, with the premises of the structural balance theory. This thesis would not have been possible without the support of Fundación BBVA through its 2016 call project ”Los números de Dunbar y la estructura de las sociedades digitales: modelización y simulación (DUNDIG)”, and we are very thankful for it. Support for early stages of this work through projects IBSEN (European Commission, H2020 FET Open RIA 662725) and VARIANCE (Ministerio de Economía y Competitividad/FEDER, project no. FIS2015-64349-P) is also acknowledged Programa Oficial de Doctorado en Ingeniería Matemática por la Universidad Carlos III de Madrid Presidente: Javier Martín Buldú.- Secretario: José Luis Molina González.- Vocal: Roberta Sinatra

After an introductory discussion, in which the main notions and background materials of discrete multiorthogonality are addressed, we focus our attention on some new results, namely Chapters 2, 3, and 4. These three chapters constitute the main core of the Thesis. Indeed, Chapter 2 is focused on the study of four new families of q-multiple orthogonal polynomials, namely q-multiple Charlier, q-multiple Meixner of the first and second kind, respectively, and q-multiple Kravchuk. The raising operators and Rodrigues-type formulas, which provide an explicit expression for these new families, are obtained. Chapter 3 contains a detailed study of some algebraic properties for the aforementioned q-families of multiple orthogonal polynomials. More specifically, the (r + 1) order recurrence relation as well as the (r + 1) order difference equations in the discrete variable on the real line are obtained. Here the letter r is used to denote the dimension of the vector measure µ [vector] . Finally, in Chapter 4 some limit relations between the attained q-families of multiple orthogonal polynomials (when the parameter q approaches 1) and discrete multiple orthogonal polynomials are established. Programa Oficial de Doctorado en Ingeniería Matemática Presidente: Francisco José Marcellán Español.- Secretario: Alejandro Zarzo Altarejos.- Vocal: Manuel Enrique Mañas Baena

Mención Internacional en el título de doctor In this Thesis we study the extremal problems of maximazing and minimazing the hyperbolicity constant on several families of graphs. In order to properly raise our research problem, we need to introduce some important definitions and make some remarks on the graphs we study. Programa Oficial de Doctorado en Ingeniería Matemática Presidente: Elena Romera Colmenarejo.- Secretario: Ana María Portilla Ferreira.- Vocal: José María Sigarreta Almira

Mención Internacional en el título de doctor We consider online task computing environments such as volunteer computing platforms running on BOINC (e.g., SETI@home) and crowdsourcing platforms such as Amazon Mechanical Turk. We model the computations as an Internet-based task computing system under the masterworker paradigm. A master entity sends tasks across the Internet, to worker entities willing to perform a computational task. Workers execute the tasks, and report back the results, completing the computational round. Unfortunately, workers are untrustworthy and might report an incorrect result. Thus, the first research question we answer in this work is how to design a reliable masterworker task computing system. We capture the workers’ behavior through two realistic models: (1) the “error probability model” which assumes the presence of altruistic workers willing to provide correct results and the presence of troll workers aiming at providing random incorrect results. Both types of workers suffer from an error probability altering their intended response. (2) The “rationality model” which assumes the presence of altruistic workers, always reporting a correct result, the presence of malicious workers always reporting an incorrect result, and the presence of rational workers following a strategy that will maximize their utility (benefit). The rational workers can choose among two strategies: either be honest and report a correct result, or cheat and report an incorrect result. Our two modeling assumptions on the workers’ behavior are supported by an experimental evaluation we have performed on Amazon Mechanical Turk. Given the error probability model, we evaluate two reliability techniques: (1) “voting” and (2) “auditing” in terms of task assignments required and time invested for computing correctly a set of tasks with high probability. Considering the rationality model, we take an evolutionary game theoretic approach and we design mechanisms that eventually achieve a reliable computational platform where the master receives the correct task result with probability one and with minimal auditing cost. The designed mechanisms provide incentives to the rational workers, reinforcing their strategy to a correct behavior, while they are complemented by four reputation schemes that cope with malice. Finally, we also design a mechanism that deals with unresponsive workers by keeping a reputation related to the workers’ response rate. The designed mechanism selects the most reliable and active workers in each computational round. Simulations, among other, depict the trade-off between the master’s cost and the time the system needs to reach a state where the master always receives the correct task result. The second research question we answer in this work concerns the fair and efficient distribution of workers among the masters over multiple computational rounds. Masters with similar tasks are competing for the same set of workers at each computational round. Workers must be assigned to the masters in a fair manner; when the master values a worker’s contribution the most. We consider that a master might have a strategic behavior, declaring a dishonest valuation on a worker in each round, in an attempt to increase its benefit. This strategic behavior from the side of the masters might lead to unfair and inefficient assignments of workers. Applying renown auction mechanisms to solve the problem at hand can be infeasible since monetary payments are required on the side of the masters. Hence, we present an alternative mechanism for fair and efficient distribution of the workers in the presence of strategic masters, without the use of monetary incentives. We show analytically that our designed mechanism guarantees fairness, is socially efficient, and is truthful. Simulations favourably compare our designed mechanism with two benchmark auction mechanisms. This work has been supported by IMDEA Networks Institute and the Spanish Ministry of Education grant FPU2013-03792. Programa Oficial de Doctorado en Ingeniería Matemática Presidente: Alberto Tarable.- Secretario: José Antonio Cuesta Ruiz.- Vocal: Juan Julián Merelo Guervós

Hyperbolic spaces, defined by Gromov in, play an important role in geometric group theory and in the geometry of negatively curved spaces. The concept of Gromov hyperbolicity grasps the essence of negatively curved spaces like the classical hyperbolic space, Riemannian manifolds of negative sectional curvature bounded away from 0, and of discrete spaces like trees and the Cayley graphs of many finitely generated groups. It is remarkable that a simple concept leads to such a rich general theory. The first works on Gromov hyperbolic spaces deal with finitely generated groups. Initially, Gromov spaces were applied to the study of automatic groups in the science of computation; indeed, hyperbolic groups are strongly geodesically automatic, i.e., there is an automatic structure on the group. The concept of hyperbolicity appears also in discrete mathematics, algorithms and networking. For example, it has been shown empirically in that the internet topology embeds with better accuracy into a hyperbolic space than into an Euclidean space of comparable dimension; furthermore, it is evidenced that many real networks are hyperbolic. A few algorithmic problems in hyperbolic spaces and hyperbolic graphs have been considered in recent papers. Another important application of these spaces is the study of the spread of viruses through the internet. Furthermore, hyperbolic spaces are useful in secure transmission of information on the network. The hyperbolicity has also been used extensively in the context of random graphs. For example, it was shown that several types of small-world networks and networks with given expected degrees are not hyperbolic in some sense. The study of Gromov hyperbolic graphs is a subject of increasing interest in graph theory; and the references therein. In our study on the hyperbolicity in graphs we use the notations (...) Programa Oficial de Doctorado en Ingeniería Matemática Presidente: Domingo de Guzmán Pestana Galván.- Secretario: Ana María Portilla Ferreira.- Vocal: Eva Touris Lojo

The first and second Frobenius companion matrices appear frequently in numerical application, but it is well known that they possess many properties that are undesirable numerically, which limit their use in applications. Fiedler companion matrices, or Fiedler matrices for brevity, introduced in 2003, is a family of matrices which includes the two Frobenius matrices. The main goal of this work is to study whether or not Fiedler companion matrices can be used with more reliability than the Frobenius ones in the numerical applications where Frobenius matrices are used. For this reason, in this work we present a thorough study of Fiedler matrices: their structure and numerical properties, where we mean by numerical properties those properties that are interesting for applying these matrices in numerical computations, and some of their applications in the field on numerical linear algebra. The introduction of Fiedler companion matrices is an example of a simple idea that has been very influential in the development of several lines of research in the numerical linear algebra field. This family of matrices has important connections with a number of topics of current interest, including: polynomial root finding algorithms, linearizations of matrix polynomials, unitary Hessenberg matrices, CMV matrices, Green’s matrices, orthogonal polynomials, rank structured matrices, quasiseparable and semiseparable matrices, etc., Programa Oficial de Doctorado en Ingeniería Matemática, Presidente: Paul Van Dooren.- Secretario: Juan Bernardo Zaballa Tejada.- Vocal: Françoise Tisseur

Programa Oficial de Doctorado en Ingeniería Matemática, Presidente: José María Arrieta Algarra.- Secretario: José Manuel Rodríguez García.- Vocal: Ana Rodríguez Santa María

Mención Internacional en el título de doctor, Cooperation and coordination is a desirable behavior that is fundamental for the harmonious development of society. People need to rely on cooperation with other individuals in many aspects of everyday life, such as teamwork and economic exchange in anonymous markets. However, cooperation may easily fall prey to exploitation by selfish individuals who only care about short-term gain. For cooperation to evolve, specific conditions and mechanisms are required, such as kinship, direct and indirect reciprocity through repeated interactions, or external interventions such as punishment. In this dissertation we investigate the effect of the network structure of the population on the evolution of cooperation and coordination. We consider several kinds of static and dynamical network topologies, such as Barabási–Albert, social network models and spatial networks. We perform numerical simulations and laboratory experiments using the Prisoner’s Dilemma and coordination games in order to contrast human behavior with theoretical results. We show by numerical simulations that even a moderate amount of random noise on the Barabási–Albert scale-free network links causes a significant loss of cooperation, to the point that cooperation almost vanishes altogether in the Prisoner’s Dilemma when the noise rate is high enough. Moreover, when we consider fixed social-like networks we find that current models of social networks may allow cooperation to emerge and to be robust at least as much as in scale-free networks. In the framework of spatial networks, we investigate whether cooperation can evolve and be stable when agents move randomly or performing L´evy flights in a continuous space. We also consider discrete space adopting purposeful mobility and binary birth-death process to discover emergent cooperative patterns. The fundamental result is that cooperation may be enhanced when this migration is opportunistic or even when agents follow very simple heuristics. In the experimental laboratory, we investigate the issue of social coordination between individuals located on networks of contacts. In contrast to simulations, we find that human players dynamics do not converge to the efficient outcome more often in a social-like network than in a random network. In another experiment, we study the behavior of people who play a pure coordination game in a spatial environment in which they can move around and when changing convention is costly. We find that each convention forms homogeneous clusters and is adopted by approximately half of the individuals. When we provide them with global information, i.e. the number of subjects currently adopting one of the conventions, global consensus is reached in most, but not all, cases. Our results allow us to extract the heuristics used by the participants and to build a numerical simulation model that agrees very well with the experiments. Our findings have important implications for policymakers intending to promote specific, desired behaviors in a mobile population. Furthermore, we carry out an experiment with human subjects playing the Prisoner’s Dilemma game in a diluted grid where people are able to move around. In contrast to previous results on purposeful rewiring in relational networks, we find no noticeable effect of mobility in space on the level of cooperation. Clusters of cooperators form momentarily but in a few rounds they dissolve as cooperators at the boundaries stop tolerating being cheated upon. Our results highlight the difficulties that mobile agents have to establish a cooperative environment in a spatial setting without a device such as reputation or the possibility of retaliation. i.e. punishment. Finally, we test experimentally the evolution of cooperation in social networks taking into account a setting where we allow people to make or break links at their will. In this work we give particular attention to whether information on an individual’s actions is freely available to potential partners or not. Studying the role of information is relevant as information on other people’s actions is often not available for free: a recruiting firm may need to call a job candidate’s references, a bank may need to find out about the credit history of a new client, etc.We find that people cooperate almost fully when information on their actions is freely available to their potential partners. Cooperation is less likely, however, if people have to pay about half of what they gain from cooperating with a cooperator. Cooperation declines even further if people have to pay a cost that is almost equivalent to the gain from cooperating with a cooperator. Thus, costly information on potential neighbors’ actions can undermine the incentive to cooperate in dynamical networks., Programa Oficial de Doctorado en Ingeniería Matemática, Presidente: José Antonio Cuesta Ruiz.- Secretario: Maxi San Miguel.- Vocal: Daniele Vilone

Mención Internacional en el título de doctor, In this thesis we have studied the effect of fluctuations in two important paradigms of soft condensed matter physics, namely, Fisher fronts and thin fluid films. Our study of stochastic FKPP traveling waves broadens current knowledge on the influence of thermal noise onto the dynamics of these waves in two space dimensions. The FKPP equation describes the evolution of a stable state into an unstable one by creating a surface (a line in one dimension) that delimits the occupancy of the stable state. This 2d surface contains equipotential lines (points, in 1d) perpendicular to the velocity direction. Due to the pulled nature of the FKPP front, the noise decreases the velocity in such way that, on average, the decrease depends logarithmically on the total number of particles. Hence, even for a macroscopic number of particles, the effect of fluctuations can be observed. In two space dimensions, the stochastic front surface roughens in time with the scale-invariance properties. Indeed, we have shown through extensive simulations of the algorithm based on a special stochastic PDE solver, that the 2d sFKPP traveling wave solutions define a surface that roughens with the same scaling exponents as the KPZ equation in 1d. To further support this claim, we have calculated the distribution function of the fluctuations in surface position, and obtained very good agreement with the distribution of the KPZ surface, which is the Tracy-Widom-GOE distribution. The fact that dimensionality of the universality class is smaller than the one of the sFKPP equation lies in the weak-noise mechanism, which only affects the microscopic region of the front. In the macroscopic region, the weak noise does not produce any additional effect. Namely, above certain length-scale, the effect that fluctuations produce in the microscopic region, simply propagates back to the macroscopic one. We show that the morphology of the microscopic surface defines that of the macroscopic surface after a time related to the total number of particles, providing the same universality class for all equipotential lines….., La investigación en física estadística se ha centrado en estudiar los estados en equilibrio que alcanza un sistema físico, mientras que la dinámica fuera del equilibrio, descrita por ejemplo por las ecuaciones diferenciales, solo es conocida en los casos más simples. El desarrollo de los ordenadores ha permitido a los investigadores estudiar numéricamente esta dinámica fuera del equilibrio. Los estudios basados en simulaciones numéricas no sólo han hecho posible observar la evolución dinámica sino también, en muchos casos han contribuido en que se encontraran o mejoraran las predicciones analíticas. Sin embargo, las simulaciones han hecho posible estudiar una gran variedad de modelos, como por ejemplo, modelos de partículas, dadas las relaciones de interacción entre sí mismas, modelos explícitos, implícitos o semi-implícitos, basados en soluciones numéricas de las ecuaciones parciales diferenciales, modelos espectrales o pseudo-espectrales, etc. Estos modelos matemáticos, aparte de modelar los sistemas físicos tienen muchas aplicaciones en otros campos, especialmente en biología, química y economía. Además, la necesidad industrial de entender cuantitativamente los procesos en escalas muy pequeñas (micro y nano) requiere estudios numéricos para ver qué modelo describe mejor dichos procesos. Los modelos en los que se centran los trabajos presentados en esta tesis describen sistemas microfluídicos, como por ejemplo la evolución de una superficie o intercara entre fluido y vapor sobre un sustrato plano. Normalmente, en esas escalas (micro y nanoscópicas) es imposible desarrollar modelos teniendo en cuenta todas las interacciones conocidas, mientras los modelos fenomenológicos a veces son demasiado simples. Por esta razón, hemos estudiado la evolución de dos modelos simples con un ruido específico que trata de agrupar los efectos que se pierden simplificando la naturaleza de las interacciones, creando modelos más realistas. El primer modelo es un proceso reactivo-difusivo dado por la ecuación de Fisher-Kolmogorov-Petrovsky-Piscounov (FKPP), utilizada recientemente en bastantes sistemas biológicos. El segundo modelo representa los sistemas de películas delgadas y la evolución de las películas bajo diferentes situaciones físicas de la interacción fluido-sustrato…., Programa Oficial de Doctorado en Ingeniería Matemática, Presidente: Aurora Hernández Machado.- Secretario: Olivier Pierre Louis.- Vocal: Javier Manuel Muñoz García

One of the open problems in graph theory is the characterization of any graph by a polynomial. Research in this area has been largely driven by the advantages offered by the use of computers which make working with graphs: it is simpler to represent a graph by a polynomial (a vector) that by the adjacency matrix (a matrix). We introduce the alliance polynomial of a graph. The alliance polynomial of a graph G with order n and maximum degree δ_1 is the polynomial A(G; x) = ∑_(k=〖-δ〗_1)^(δ_1)▒〖Ak(G) x^(n+k) 〗, where A{_k}(G) is the number of exact defensive k-alliances in G. Also, we develop and implement an algorithm that computes in an efficient way the alliance polynomial. We obtain some properties of A(G; x) and its coefficients for: • Path, cycle, complete and star graphs. In particular, we prove that they are characterized by their alliance polynomials. • Cubic graphs (graphs with all of their vertices of degree 3), since they are a very interesting class of graphs with many applications. We prove that they verify unimodality. Also, we compute the alliance polynomial for cubic graphs of small order, which satisfy uniqueness. • Regular graphs (graphs with the same degree for all vertices). In particular, we characterize the degree of regular graphs by the number of non-zero coefficients of their alliance polynomial. Besides, we prove that the family of alliance polynomials of connected ∆-regular graphs with small degree is a very special one, since it does not contain alliance polynomials of graphs which are not connected ∆-regular. If X is a geodesic metric space and x1, x2, x3 ∈ X, a geodesic triangle T = {x1, x2, x3} is the union of the three geodesics [x1x2], [x2x3] and [x3x1] in X. The space X is δ-hyperbolic (in the Gromov sense) if any side of T is contained in the δ-neighborhood of the union of the two other sides, for every geodesic triangle T in X. We denote by δ(X) the sharp hyperbolicity constant of X, i.e., δ(X) := inf{δ >= 0 : X is δ-hyperbolic }. The study of hyperbolic graphs is an interesting topic since the hyperbolicity of a geodesic metric space is equivalent to the hyperbolicity of a graph related to it. We obtain information about the hyperbolicity constant of cubic graphs. These graphs are also very important in the study of Gromov hyperbolicity, since for any graph G with bounded maximum degree there exists a cubic graph G* such that G is the hyperbolic if and only if G* is hyperbolic. We find some characterizations for the cubic graphs which have small hyperbolicity constants. Besides, we obtain bounds for the hyperbolicity constant of the complement graph of a cubic graph; our main result of this kind says that for any finite cubic graph G which is not isomorphic either to K_4 or to K_3,3, the inequalities 5k/4, Uno de los problemas abiertos en la teoría de grafos es la caracterización de cualquier grafo por un polinomio. La investigación en este área ha sido impulsada en gran parte por las ventajas que ofrece el uso de las computadoras que hacen que trabajar con grafos sea más simple. En esta Tesis introducimos el polinomio de alianza de un grafo. El polinomio de alianza de un grafo G con orden n y grado m´aximo δ_1 es el polinomio A(G; x) = ∑_(k=〖-δ〗_1)^(δ_1)▒〖Ak(G) x^(n+k) , donde A{_k}(G) es el n´umero de k alianzas defensivas exactas en G. También desarrollamos e implementamos un algoritmo que calcula de manera eficiente el polinomio de alianza. En este trabajo obtenemos algunas propiedades de A(G; x) y sus coeficientes para: • Grafos caminos, ciclos, completos y estrellas. En particular, hemos demostrado que se caracterizan mediante sus polinomios de alianza. • Grafos cúbicos (grafos con todos sus vértices de grado 3), ya que son una clase muy interesante de grafos con muchas aplicaciones. Hemos demostrado que sus polinomios de alianza verifican unimodalidad. Además, calculamos el polinomio de alianza para grafos cúbicos de orden pequeño, los cuales satisfacen unicidad. • Grafos regulares (grafos con todos sus vértices de igual grado). En particular, se caracteriza el grado de los grafos regulares por el n´umero de coeficientes distintos de cero de su polinomio de alianza. Además, se demuestra que la familia de polinomios de alianza de grafos conexos _-regulares con grado pequeño es muy especial, ya que no contiene polinomios de alianza de grafos conexos que no sean _-regulares. Si X es un espacio métrico geodésico y x1, x2, x3 ∈ X, un triángulo geodésico T = {x1, x2, x3} es la unión de tres geodésicas [x1x2], [x2x3] and [x3x1] de X. El espacio X es δ-hiperbólico (en el sentido de Gromov) si todo lado de todo triángulo geodésico T de X está contenido en la δ-vecindad de la unóon de los otros dos lados. Se denota por δ(X) la constante de hiperbolicidad óptima de X, es decir, δ(X) := inf{δ > 0 : X es δ-hiperbólico }. El estudio de los grafos hiperbólicos es un tema interesante dado que la hiperbolicidad de un espacio métrico geodésico es equivalente a la hiperbolicidad de un grafo más sencillo asociado al espacio. Hemos obtenido información acerca de la constante de hiperbolicidad de los grafos cúbicos; dichos grafos son muy importantes en el estudio de la hiperbolicidad, ya que para cualquier grafo G con grado máximo acotado existe un grafo cúbico G∗ tal que G es hiperbólico si y sólo si G∗ es hiperbólico. En esta memoria conseguimos caracterizar los grafos cúbicos con constante de hiperbolicidad pequeña. Además, se obtienen cotas para la constante de hiperbolicidad del grafo complemento de un grafo cúbico; nuestro principal resultado dice que para cualquier grafo cúbico finito G no isomorfo a K4 o K3,3, se cumple la relación 5k/4, Programa de Doctorado en Ingeniería Matemática por la Universidad Carlos III de Madrid, Presidente: Domingo de Guzmán Pestana Galván; Secretario: Eva Tourís Lojo; Vocal: Sergio Bermudo Navarrete

If X is a geodesic metric space and x1, x2, x3 ∈ X, a geodesic triangle T = {x1, x2, x3} is the union of the three geodesics [x1x2], [x2x3] and [x3x1] in X. The space X is δ-hyperbolic (in the Gromov sense) if any side of T is contained in the δ-neighborhood of the union of the two other sides, for every geodesic triangle T in X. We denote by δ(X) the sharp hyperbolicity constant of X, i.e., δ(X) := inf{δ ≥ 0 : X is δ-hyperbolic } . The study of hyperbolic graphs is an interesting topic since the hyperbolicity of a geodesic metric space is equivalent to the hyperbolicity of a graph related to it. One of the main aims of this PhD Thesis is to obtain quantitative information about the distortion of the hyperbolicity constant of the graph G \ e obtained from the graph G by deleting an arbitrary edge e from it. These inequalities allow to obtain other main result, which characterizes in a quantitative way the hyperbolicity of any graph in terms of local hyperbolicity. In this work we also obtain information about the hyperbolicity constant of the line graph L(G) in terms of properties of the graph G. In particular, we prove qualitative results as the following: a graph G is hyperbolic if and only if L(G) is hyperbolic; if {Gn} is a T-decomposition of G ({Gn} are simple subgraphs of G), the line graph L(G) is hyperbolic if and only if supn δ(L(Gn)) is finite. Besides, we obtain quantitative results when k is the length of the edges of G and L(G). Two of them are quantitative versions of our qualitative results. We also prove that g(G)/4 ≤ δ(L(G)) ≤ c(G)/4 + 2k, where g(G) is the girth of G and c(G) is its circumference. We show that δ(L(G)) ≥ sup{L(g) : g is an isometric cycle in G}/4. Besides, we obtain bounds for δ(G) + δ(L(G)). Also, we characterize the graphs G with δ(L(G)) < k. Furthermore, we consider G with edges of arbitrary lengths, and L(G) with edges of non-constant lengths. In particular, we prove that a cycle of the graph G is transformed isometrically into a cycle of the graph L(G) with the same length. We also prove that δ(G) ≤ δ(L(G)) ≤ 5δ(G) + 3lmax, where lmax := supe∈E(G) L(e). This result implies the monotony of the hyperbolicity constant under a non-trivial transformation (the line graph of a graph). Also, we obtain criteria which allow us to decide, for a large class of graphs, whether they are hyperbolic or not. We are especially interested in the planar graphs which are the “boundary” (the 1-skeleton) of a tessellation of the Euclidean plane. Furthermore, we prove that a graph obtained as the 1-skeleton of a general CW 2-complex is hyperbolic if and only if its dual graph is hyperbolic. One of the main problems on this subject is to relate the hyperbolicity with other properties on graph theory. We extend in two ways (edge-chordality and path-chordality) the classical definition of chordal graphs in order to relate this property with Gromov hyperbolicity. In fact, we prove that every edge-chordal graph is hyperbolic and that every hyperbolic graph is path-chordal. Furthermore, we prove that every path-chordal cubic graph (with small path-chordality constant) is hyperbolic. Some previous works characterize the hyperbolic product graphs (for the Cartesian product, strong product and lexicographic product) in terms of properties of the factor graphs. Finally, we characterize the hyperbolic product graphs for two important kinds of products: the graph join G1 ⊎G2 and the corona G1 ⋄G2. The graph join G1 ⊎G2 is always hyperbolic, and G1 ⋄ G2 is hyperbolic if and only if G1 is hyperbolic. Furthermore, we obtain simple formulae for the hyperbolicity constant of the graph join G1 ⊎ G2 and the corona G1 ⋄ G2., Sea X un espacio métrico geodésico y x1, x2, x3 ∈ X. Un triángulo geodésico T = {x1, x2, x3} es la unión de tres geodésicas [x1x2], [x2x3] y [x3x1] de X. El espacio X es δ- hiperbólico (en el sentido de Gromov) si todo lado de T está contenido en la δ-vecindad de la unión de los otros dos lados, para todo triángulo geodésico T de X. Se denota por δ(X) a la constante de hiperbolicidad óptima de X, es decir, δ(X) := inf{δ ≥ 0 : X es δ-hiperbólico }. El estudio de los grafos hiperbólicos es un tema interesante dado que la hiperbolicidad de un espacio métrico geodésico es equivalente a la hiperbolicidad de un grafo más sencillo asociado al espacio. Uno de los principales objetivos de esta Tesis Doctoral es obtener información cuantitativa sobre la variación de la constante de hiperbolicidad del grafo G \ e que se obtiene del grafo G mediante la eliminación de una arista arbitraria e de él. Estas desigualdades permiten caracterizar, de forma cuantitativa, la hiperbolicidad de cualquier grafo en términos de su hiperbolicidad local. En esta memoria se obtene información acerca de la constante de hiperbolicidad del grafo línea L(G) en términos de propiedades del grafo G. En particular, se obtienen los siguientes resultados cualitativos: un grafo G es hiperbólico si y sólo si L(G) es hiperbólico; si {Gn} es una T-descomposición de G, el grafo línea L(G) es hiperbólico si y sólo si supn δ(L(Gn)) es finito. Además, se obtienen resultados cuantitativos cuando las aristas de G y L(G) tienen longitud k. Se demuestra que g(G)/4 ≤ δ(L(G)) ≤ c(G)/4 + 2k, donde g(G) es el cuello de G y c(G) es su circunferencia. También se prueba que δ(L(G)) ≥ sup{L(g) : g es un ciclo isométrico de G}/4. Igualmente, se obtienen cotas para δ(G) + δ(L(G)) y se caracterizan los grafos G tales que δ(L(G)) < k. Por otra parte, se consideran grafos G con aristas de longitudes arbitrarias, y L(G) con aristas de longitudes no constante. En particular, se demuestra que los ciclos de G se transforman isométricamente en ciclos de L(G) con la misma longitud. También se obtiene la relación δ(G) ≤ δ(L(G)) ≤ 5δ(G) + 3lmax, donde lmax := supe∈E(G) L(e). Este resultado prueba la monotonía de la constante de hiperbolicidad bajo una transformación no trivial (el grafo línea de un grafo). También, se obtienen criterios que permiten decidir, para una clase amplia de grafos, cuando son estos hiperbólicos o no. Se presta especial atención en los grafos planares que son la “frontera” (el 1-esqueleto) de una teselación del plano euclídeo. Además, se prueba que los grafos que se obtienen como 1-esqueleto de un CW 2-complejo general son hiperbólicos si y sólo si su grafo dual es hiperbólico. Uno de los principales problemas en este área es relacionar la hiperbolicidad de un grafo con otras propiedades de la teoría de grafos. En este trabajo se extiende de dos maneras (arista-cordalidad y camino-cordalidad) la definición clásica de cordalidad con el fin de relacionar esta propiedad con la hiperbolicidad. De hecho, se demuestra que todo grafo arista-cordal es hiperbólico y que todo grafo hiperbólico es camino-cordal. También, se demuestra que todo grafo cúbico camino-cordal (con constante pequeña de camino-cordalidad) es hiperbólico. Algunos trabajos previos caracterizan la hiperbolicidad de productos de grafos (para el producto cartesiano, el producto fuerte y el producto lexicográfico) en términos de propiedades de los grafos factores. En el último capítulo, se caracteriza la hiperbolicidad de dos productos de grafos: el grafo join G1 ⊎ G2 y el corona G1 ⋄ G2. El grafo join G1 ⊎ G2 siempre es hiperbólico, y el corona G1 ⋄ G2 es hiperbólico si y sólo si G1 es hiperbólico. Además, obtenemos fórmulas sencillas para la constante de hiperbolicidad del grafo join G1 ⊎ G2 y del corona G1 ⋄ G2.

Mención Internacional en el título de doctor, The study of interfacial phenomena has always constituted an integral part of condensed matter physics and materials science. Indeed, most properties of real materials depend crucially on the presence of imperfections, such as bulk vacancies, dislocations, surface roughness, etc. which derive from the non-equilibrium conditions under which the material has formed [1, 2]. In the last century, the increasing interest in systems with considerable surface to volume ratio, such as for instance devices at the nanoscale, has attracted scientists from different fields, from physics to chemistry, biology, or engineering [3]. This is due to the great amount of technological applications of such systems to a wide variety of situations. Moreover, the improvement of production and characterization techniques for the growth of surfaces at micro and nano-scale, such as Molecular Beam Epitaxy, or for surface etching, such as Ion Beam Sputtering , have unveiled unexpected interesting physical properties of the grown interfaces. From a technological point of view, the ability to control and predict the effect induced by disorder and the mechanisms of self-organization that ensue during the growth dynamics is of great interest [4]. For instance, the possibility to control the surface roughness could improve electric conductivity or the mechanical contact of certain devices, whereas the ability to control the formation of a pattern could change, for instance, the optical properties of the material., Esta tesis se centra en fenómenos de invariancia de escala genérica en modelos de crecimiento de superficies. Concretamente, hemos considerado sistemas fuera de equilibrio que presentaran desorden a grandes escalas espaciales y temporales (rugosidad cinética), y que estuvieran descritos por ecuaciones continuas. El objetivo de la tesis ha sido utilizar métodos propios de la mecánica estadística para estudiar la interacción de este tipo de sistemas con otros dos fenómenos: la aparición de inestabilidades morfológicas que dan lugar a la formación de patrones, y la presencia de anisotropía. En cuanto al primer problema, hemos considerado un ejemplo paradigmático de ecuación que presente invariancia de escala genérica e inestabilidades morfológicas, la ecuación de Kuramoto-Siashinsky. Hasta la fecha, el comportamiento asintótico de este modelo no se había entendido completamente. De hecho, existe una controversia acerca de cual sea la clase de universalidad a la que este modelo pertenece en dos dimensiones. Utilizando simulaciones numéricas de gran escala, hemos comprobado que esta ecuación en dos dimensiones pertenece a la clase de universalidad de Kardar-Parisi-Zhang. Sucesivamente, nos hemos centrado en modelos que presenten invariancia de escala genérica, cuyos exponentes críticos sean distintos cuando se miden en direcciones distintas del sustrato. Cuando eso ocurre, decimos que el sistema presenta anisotropía fuerte. A pesar de la presencia generalizada de anisotropía en sistemas naturales, hasta el momento se había dedicado relativamente poca atención a este fenómeno. Eso se debe probablemente al hecho de que los modelos mas estudiados en este ´ambito no presentan anisotropía fuerte, aunque la forma de la ecuación sea en algunos casos completamente anisótropa. Después de introducir un marco teórico para el estudio de modelos fuertemente anisótropos, hemos comprobado nuestra hipótesis a través de simulaciones numéricas de distintas ecuaciones que presentaran anisotropía fuerte. Sucesivamente, hemos llevado a cabo un estudio numérico y analítico de otros modelos, con el objetivo de encontrar condiciones para la aparición de anisotropía fuerte. Encontramos que este fenómeno ocurre en presencia de una dinámica conservada, siempre que la ecuación tenga una forma bastante especifica. Finalmente, hemos presentado evidencias experimentales de la aparición de anisotropía fuerte, a través de un análisis de datos de experimentos de erosión iónica en distintas condiciones experimentales......., Programa Oficial de Doctorado en Ingeniería Matemática, Presidente: Mario Castro Ponce.- Secretario: Javier Manuel Muñoz García.- Vocal: Gnnar Pruessner

El objetivo principal de esta disertación es el estudio de la teoría de las álgebras de Lie–Jordan Banach y su papel en el marco de la mecánica clásica y cuántica y sus aplicaciones en diferentes ámbitos de las matemáticas. La investigación algebraica de los fundamentos de la física surgió de la búsqueda de una formulación “excepcional” para la mecánica cuántica, que finalmente culminó en la formulación de las álgebras de Jordan y las posteriores C∗–álgebras..........

This work presents a study of (M;N)-coherent pairs of order (m; k) of sequences of orthogonal polynomials of a continuous and discrete variable on the real line and on the unit circle. This concept extends all the generalizations of the notion of, in our terminology, (1; 0)-coherent pair, studied in the literature, which was first introduced as coherent pair by A. Iserles, P. E. Koch, S. P. N rsett, and J. M. Sanz-Serna in 1991 (see [48])..... ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ Este trabajo presenta un estudio de los pares (M;N)-coherentes de orden (m; k) de sucesiones de polinomios ortogonales de una variable continua en la recta real y en la circunferencia unidad, y de una variable discreta. Este concepto extiende todas las generalizaciones de la noción de, en nuestra terminología, par (1,0)-coherente, estudiadas en la literatura, el cual fue introducido por primera vez como par coherente por A. Iserles, P. E. Koch, S. P. N rsett, y J. M. Sanz-Serna en 1991 (véase [48])......

This thesis deals with the mathematical modeling of evolutionary processes that take place in heterogeneous populations. Its leitmotif is the response of complex ensembles of replicating entities to multiple (and often opposite) selection pressures. Even though the specific problems addressed in different chapters belong to different organizational levels—genome, population, and community—all of them can be conceptualized as the evolution of a heterogeneous population—let it be a population of genomic elements, viruses, or prokaryotic hosts and phages—facing a complex environment. As a result, the mathematical tools required for their study are quite similar. In contrast, the strategies that each population has discovered to perpetuate vary according to the different evolutionary challenges and environmental constraints that the population experiences. Along this thesis, there has been a special interest on connecting theoretical models with experimental results. To that end, most of the work presented here has been motivated either by laboratory findings or by the bioinformatic analysis of sequenced genomes. We strongly believe that such a multidisciplinary approach is necessary in order to improve our knowledge on how evolution works. Moreover, experiments are a must when it comes to propose antiviral strategies based on theoretical predictions, as we do in Chapter 3. This thesis is structured in two main blocks. The first one focuses on studying instances of viral evolution under the action of mutagenic drugs, paying particular attention to their possible application to the development of novel antiviral therapies. This block comprises chapters 2 and 3; the former dicussing the phenomenon of lethal defection and stochastic viral extinction; the latter dealing with the optimal way to combine mutagens and inhibitors in multidrug antiviral treatments. The second block is devoted to the study of the evolutionary forces underlying genome structure. In chapter 4, we propose a mechanism through which multipartite viruses could have originated. Interestingly, the pathway leading to genome segmentation shares some steps with lethal defection, but each outcome is reached at specific environmental conditions. Chapter 5 analyses the abundance distributions of transposable elements in prokaryotic genomes, with the aim of determining the key processes involved in their spreading. We explicitly explore the hypothesis that transposable elements follow a neutral dynamics, with a negligible fitness cost for their host genomes. A higher level of organization is studied in Chapter 6, where an agent based coevolutionary model based on Lotka-Volterra interactions is used to investigate the evolutionary dynamics of the prokaryotic antiviral immunity system CRISPR-Cas. This chapter also examines the environmental factors that are responsible for its maintenance or loss. Finally, Chapter 7 summarizes the main results obtained along the thesis and sketches possible lines of work based on them.

Esta tesis incluye cinco capítulos, y está dedicada al estudio de familias de polinomios ortogonales de tipo estándar y no estándar. A continuación resumimos brevemente las conclusiones aportadas por la presente memoria. • Se realiza por primera vez un estudio completo del comportamiento de los zeros de familias de polinomios ortogonales con respecto a una medidas modificadas mediante perturbaciones canónicas de tipo Uvarov y Christoffel . El comportamiento de estos ceros se da en términos del parámetro M, el cual determina cómo es la intensidad de la perturbación sobre la medida clásica. Hasta el momento, se habían realizado progresos significativos en esta dirección a través de aproximaciones semiclásicas, y solamente tratando el comportamiento de propiedades promedio de los ceros, usando el método de WKB. • Se obtienen resultados asintóticos de las secuencias de polinomios ortogonales mónicos con respecto a la perturbación de Uvarov de la medida clásica de Laguerre, como ejemplo canónico de perturbación fuera del soporte de una medida clásica con soporte no acotado. Hasta la fecha, los puntos de masa se localizaban en la frontera (o fronteras) del soporte de la medida perturbada. • Se proporciona un modelo electrostático de los ceros de la familia de polinomios ortogonales con respecto a una medida de Laguerre perturbada, con una cantidad numerable m de puntos de masa, fuera del soporte de la medida clásica de Laguerre. Hasta el momento, el único trabajo similar consideraba un solo punto de masa en el origen. Igualmente, describimos el comportamiento de los ceros de los polinomios ortogonales tipo-Krall en términos de los ceros de cierto polinomio de grado 2m (siendo m el número de masas de Dirac que aparecen en la medida), y que son las fuentes de un potencial logarítmico de corto alcance que afecta a la localización de los ceros de las secuencias de polinomios ortogonales Krall-Laguerre, considerados como puntos críticos de un problema de equilibrio. • Igualmente se obtienen propiedades asintóticas de secuencias de polinomios ortogonales mónicos de tipo Laguerre-Sobolev, cuando los puntos de masa están situados fuera del soporte de la medida clásica de Laguerre. Hasta la fecha, los puntos de masa se localizaban en la frontera (o fronteras) de los soportes de las medidas perturbadas.

The main purpose of the work presented here is to study transformations of sequences of orthogonal polynomials associated with a hermitian linear functional L, using spectral transformations of the corresponding C-function F. We show that a rational spectral transformation of F is given by a finite composition of four canonical spectral transformations. In addition to the canonical spectral transformations, we deal with two new examples of linear spectral transformations. First, we analyze a spectral transformation of L such that the corresponding moment matrix is the result of the addition of a constant on the main diagonal or on two symmetric sub-diagonals of the initial moment matrix. Next, we introduce a spectral transformation of L by the addition of the first derivative of a complex Dirac linear functional when its support is a point on the unit circle or two points symmetric with respect to the unit circle. In this case, outer relative asymptotics for the new sequences of orthogonal polynomials in terms of the original ones are obtained. Necessary and su cient conditions for the quasi-definiteness of the new linear functionals are given. The relation between the corresponding sequence of orthogonal polynomials in terms of the original one is presented. We also consider polynomials which satisfy the same recurrence relation as the polynomials orthogonal with respect to the linear functional L , with the restriction that the Verblunsky coe cients are in modulus greater than one. With positive or alternating positive-negative values for Verblunsky coe cients, zeros, quadrature rules, integral representation, and associated moment problem are analyzed. We also investigate the location, monotonicity, and asymptotics of the zeros of polynomials orthogonal with respect to a discrete Sobolev inner product for measures supported on the real line and on the unit circle., El objetivo principal de este trabajo es el estudio de las sucesiones de polinomios ortogonales con respecto a transformaciones de un funcional lineal hermitiano L , usando para ello las transformaciones de la correspondiente C-función F . Un primer resultado es que las transformaciones espectrales racionales de F están dadas por una composición finita de cuatro transformaciones espectrales canónicas. Además de estas transformaciones canónicas se estudian dos ejemplos de transformaciones espectrales lineales que son novedosos en la literatura. El primero de estos ejemplos está dado por una modificación del funcional lineal L, de modo que la correspondiente matriz de momentos es el resultado de la adición de una constante en la diagonal principal o en dos subdiagonales simétricas de la matriz de momentos original. El segundo ejemplo es una transformación de L mediante la adición de la primera derivada de una delta de Dirac compleja cuando su soporte es un punto sobre la circunferencia unidad o dos puntos simétricos respecto a la circunferencia unidad. En este caso se obtiene la asintótica relativa exterior de la nueva sucesión de polinomios ortogonales en términos de la original. Se dan condiciones necesarias y suficientes para que los funcionales derivados de las perturbaciones estudiadas sean cuasi-definidos, y se obtiene la relación entre las correspondientes sucesiones de polinomios ortogonales. Se consideran además polinomios que satisfacen las mismas ecuaciones de recurrencia que los polinomios ortogonales con respecto al funcional lineal L, agregando la restricción de que sus coeficientes de Verblunsky son en valor absoluto mayores que 1. Cuando estos coeficientes son positivos o alternan signo, se estudian los ceros, las fórmulas de cuadratura, la representación integral y el problema de momentos asociado. Asimismo, se estudia la localización, monotonicidad y comportamiento asintótico de los ceros asociados a polinomios discretos ortogonales de Sobolev para medidas soportadas tanto en la recta real como en la circunferencia unidad., This work was supported by FPU Research Fellowships, Re. AP2008-00471 and Dirección General de Investigación, Ministerio de Ciencia e Inovación of Spain, grant MTM2009-12740-C03-01.

Programa de Doctorado en Ingeniería Matemática por la Universidad Carlos III de Madrid Presidente: Luis Alberto Ibort Latre.- Secretario: Froilán César Martínez Dopico. - Vocales: Juan Carlos Marreno González, Cantrijn Frans, Narciso Román Roy